This work proposes neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models and establishes a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory.
Abstract
Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial conditions differ. For linear state space models, such control terms can be derived under mild assumptions. However, designing effective nudging terms becomes significantly more challenging in the nonlinear setting. In this work, we propose neural network nudging, a data-driven method for learning nudging terms in nonlinear state space models. We establish a theoretical existence result based on the Kazantzis--Kravaris--Luenberger observer theory. The proposed approach is evaluated on three benchmark problems that exhibit chaotic behavior: the Lorenz 96 model, the Kuramoto--Sivashinsky equation, and the Kolmogorov flow.
A multi-regime RC framework in which multiple readouts are trained under different dynamical conditions and combined through a short observation window to form a trajectory-dependent linear readout enables both regime identification and adaptation to unseen or intermediate dynamics.
S. Hadipour Lakmesari, H. Kantz, Francesco Sorrentino· Chaos· 0 citations
Multi-step training of sparse, interpretable models of dynamical systems directly from time-series data yields models with accurate short-term dynamics and strong agreement in long-time statistical properties, including mean, variance, and Lyapunov exponents.
In this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.
Edoardo Caldarelli, Oleksii Kachaiev, C. Molinari et al.· 0 citations
An eigenanalysis framework is developed that reveals the dynamical origin of inference-time error growth and introduces a stability-promoting loss that explicitly regularizes Jacobian-driven error amplification, improving both forecast accuracy and dynamical robustness.
C. Ainslie, P. Hassanzadeh, Michael W. Mahoney et al.· 0 citations
The combination of analytical and computational perspectives provides a clear framework for understanding this generalized equation and offers a practical approach for investigating other nonlinear systems with a similar structure.
Muhammad Ghulam Abbas Malik, Muhammad Mudassir, Zia Bashir· Mathematical and Computation...· 0 citations
This letter develops a data-driven control framework for nonlinear ensemble systems using reservoir computing (RC). We consider ensemble control problems, in which the objective is to regulate a large, potentially uncountable, population of systems with unknown dynamics. To address this challenge, we introduce a moment kernelization approach that yields a dual representation and enables a valid finite-dimensional approximation of ensemble dynamics. Building on this reduction, we cast ensemble control synthesis as the approximation of a causal operator that maps moment trajectories to control inputs. We show that continuous-time reservoir systems induce well-defined causal input-output operators with the fading-memory property, providing a principled foundation for learning these feedback operators from moment trajectory data. Based on this theory, we design an RC-based controller trained on input-output moment trajectories and deployed in a closed-loop configuration for tracking and stabilization of nonlinear ensemble systems.
Yuan-Hung Kuan, Lin Tang, Jr-Shin Li· IEEE Control Systems Letters· 0 citations
A new machine-learning framework aims to improve the success rate of computational protein design while moving away from results that reproduce sequences found in nature.
MIT News · Artificial Intelligence· news.mit.eduAug 24, 2026
